We describe correlator product states, a class of numerically
efficient many-body wave functions to describe strongly
correlated wave functions in any dimension. Correlator product
states introduce direct correlations between physical degrees of
freedom in a simple way, yet provide the flexibility to describe
a wide variety of systems. Variational Monte Carlo calculations
for the Heisenberg and spinless fermion Hubbard models
demonstrate the promise of correlator product states for
describing both two-dimensional and fermion correlations. In one
dimension, correlator product states appear competitive with
matrix product states for the same number of variational
parameters. [Preview Abstract]

The Truncated Eigenfermion Decomposition provides a unified
computational framework for the calculation of eigenvalues,
reduced density matrices, and transition density matrices of
many-fermion Hamiltonians. Computations are made tractable by
truncating the many-fermion operator basis that is used to
approximate transformed Hamiltonians. Operator bases of
increasing size and computational complexity can be arranged in a
hierarchy that enables a systematic reduction of truncation
errors. The
suitability of this formalism for the study of strongly
correlated electrons is assessed by studying the Hubbard model on
finite clusters. Results are presented as a function of operator
basis size, cluster size, interaction strength, and doping. [Preview Abstract]

The spin and hole dynamics in a 2D hole-doped quantum
antiferromagnet
is studied for small but finite hole-doping
fractions within the
two-dimensional $t-J$ and the related
$t-t^\prime-t^{\prime\prime}-J$
model. The non-crossing approximation is used to sum up the
self-energy
diagrams and the Dyson's equations for both the hole and magnon
Green's
functions are solved self-consistently. The evolution with doping
of the hole and magnon
spectra with doping is determined, the Fermi surface
topology is
studied and the doping-dependent staggered magnetization of the
system
is computed. The latter determines the doping fraction up to which
the spin wave theory remains a
reasonable approximation to describe the doped antiferromagnet.
[Preview Abstract]

The magnetic and electronic properties of diverse systems such as
diluted, magnetic semiconductors, manganites and europium
hexaboride can be described using kinetic-exchange models in
which itinerant carriers are coupled to local magnetic moments.
Monte Carlo simulations of the magnetic properties of such models
usually treat the local moment spins as classical and ignore
electron-electron interactions due to the need to diagonalize a
fermion problem at each spin flip. We introduce a hybrid Monte
Carlo scheme that allows electron-electron interactions to be
included at a mean field level. We identify regions in parameter
space where our approach is most useful and present results of
simulations of thermodynamic quantities in ordered and disordered
models of interacting fermions coupled to local moments.
[Preview Abstract]

The matrix product state (MPS) representation used by DMRG is
extremely effective in 1D, but loses effectiveness exponentially
with the width in ladder systems. Tensor product wavefunction,
such as PEPS, are efficient representations of 2D states but
calculations utilizing them are very inefficient. As an
intermediate approach between these, we consider a wavefunction
consisting of an MPS multiplied by local bond exponentials
$^{-}\tau H_{bond}$ applied on adjacent
sites in
the lattice which appear distant in the MPS. The exponentials
restore the
area law to the MPS. For efficient calculation, the
bond-exponentials are
transferred to the Hamiltonian in a matrix product operator
representation,
acting as a similarity transformation, preserving the
eigenvalues. For this
method to be successful, not only should the modified MPS
representation be
efficient, the MPO representation of the transformed Hamiltonian
should have
a small matrix dimension. We report on preliminary results on
ladders with
several legs for this approach. [Preview Abstract]

Correlator product states are a class of many-body wave functions
that allow the
efficient numerical simulation of strongly correlated systems in
any dimension.
We have developed algorithms to approximate the time evolution of
correlator product states.
Evolution in imaginary time projects an arbitrary wave function
onto the
ground state of the system.
Real time evolution simulates the dynamics of a system and can be
used to
construct the spectral density function.
We present studies of the time evolution of correlator product
states for
the Heisenberg model in one, two, and three dimensions. [Preview Abstract]

Frustration has been a major topic in the study of
magnetism for decades. Its presence in material systems
is linked to the realization of exotic phases of matter -
spin glasses, spin ices, spin liquids. With the recent discovery
of several geometrically frustrated itinerant electron materials,
where the relevant physics occurs on a lattice constructed from
connected triangles, the role of frustration
in interacting Fermi systems has become an increasingly important
question. Numerical methods have been indispensable in the
development of our understanding of frustrated magnets.
The same will certainly hold for the study of frustrated
itinerant electrons. From this perspective, we use several
numerical techniques to study the frustrated Hubbard model.
Our initial concerns are to explore the general
physics of the frustrated Hubbard model at half-filling and
to test the effectiveness of different numerical techniques in the
presence of frustrating interactions. Particular emphasis is
placed on the application of the constrained path/phase
quantum Monte Carlo method to the frustrated Hubbard model. [Preview Abstract]

We propose an effective two-band model, which is similar to the
model considered previously by Frenkel, Gooding, Shraiman, and
Siggia (PRB 41, number 1, page 350), for describing high-T$_c$
superconducting cuprates. Instead of rotating away all oxygen
states to construct the effective sites in the Hubbard or t-J
model, we keep all oxygens allowing the hopping of the doped
holes under an antiferromagnet background on the copper sites.
This approach will be useful to explain STM experimental data
which resolves the oxygen atoms (for example, Hanaguri et al
Nature 3, 865. 2007). The magnitudes of the interactions in
two-band model are derived from a three-band Hubbard model with
reasonable parameters by applying numerical canonical
transformations with appropriate truncations. The proposed
two-band model is studied by applying DMRG calculation for
different lattices, such as ladders, and the results are compared
with same calculations done for three band model. [Preview Abstract]

This paper presents an efficient algorithm for computing the
transition probability in auxiliary field quantum Monte Carlo
simulations of strongly correlated electron systems using a
Hubbard model. This algorithm is based on a low-rank updating of
the underlying linear algebra problem, and results in significant
computational savings. The computational complexity of computing
the transition probability and Green's function update reduces to
$O(k^2)$ during the $k$-th step, where $k$ is the
number of accepted spin flips, and results in an algorithm that
is faster than the competing delayed update algorithm. Moreover,
this algorithm is orders of magnitude faster than traditional
algorithms that use naive updating of the Green's function matrix.
[Preview Abstract]

A self-consistent solution of the Hubbard model is performed on a
4x4 cluster at both the one and the two-particle level. We
combine the Parquet and the Bethe-Salpeter equations into one
non-linear equation to take advantage of optimized linear solvers
such as GMRES and BICG-Stab. We calculate some relevant
quantities and compare them to the results obtained from
Determinant Quantum Monte Carlo (DQMC), self-consistent second
order approximation and FLuctuation EXchange (FLEX)
approximation. We find that the parquet approximation, where the
fully irreducible vertex is approximated by the bare vertex,
shows satisfactory agreement with DQMC and a significant
improvement from FLEX or self-consistent second order approximation.
[Preview Abstract]

We study the phase diagram of the two-dimensional
Hubbard model in the vicinity of the quantum critical point which
separates the fermi liquid from the pseudogap region. We
use the Dynamical Cluster Approximation (DCA) in
conjunction with the weak-coupling continuous time quantum Monte
Carlo
(CTQMC) cluster solver. We measure the filling $n_c$ and the density
of states at the critical point as a function of the Coulomb
interaction $U$.
We observe a change in behavior when the Coulomb interaction is of
the order of the bandwidth. We also evaluate the temperature range in
which the system is under the influence of the quantum critical point
and compare it with the effective spin coupling $J_{eff}$. We discuss
the consistency of these results with various mechanisms of quantum
criticality. This research is supported by NSF DMR-0706379 and
OISE-0952300. [Preview Abstract]

It has been conjectured that transport in integrable one-dimensional (1D) systems is
necessarily ballistic. The large diffusive response seen
experimentally in nearly ideal realizations of the $S=1/2$ 1D Heisenberg
model is therefore puzzling and has not been explained so far. Here, we show
that, contrary to common belief, diffusion is universally present in
interacting 1D systems subject to a periodic lattice potential. We present a
parameter-free formula for the spin-lattice relaxation rate which is in
excellent agreement with experiment. Furthermore, we calculate the current
decay directly in the thermodynamic limit using a time-dependent density
matrix renormalization group algorithm and show that an anomalously large
time scale exists even at high temperatures.\\*[0.5cm]
J. Sirker, R.G. Pereira, I. Affleck, PRL (2009, in print)
[Preview Abstract]